On discrete subgroups containing a lattice in a horospherical subgroup

We showed in [Oh] that for a simple real Lie groupG with real rank at least 2, if a discrete subgroup Γ ofG contains lattices in two opposite horospherical subgroups, then Γ must be a non-uniform arithmetic lattice inG, under some additional assumptions on the horospherical subgroups. Somewhat surprisingly, a similar result is true even if we only assume that Γ contains a lattice in one horospherical subgroup, provided Γ is Zariski dense inG.     

Conjecture of Li and Praeger concerning the isomorphisms of Cayley graphs of A5

LetG be a finite group, andS a subset ofG \ |1| withS =S ⁻¹. We useX = Cay(G,S) to denote the Cayley graph ofG with respect toS. We callS a Cl-subset ofG, if for any isomorphism Cay(G,S) ≈ Cay(G,T) there is an α∈ Aut(G) such thatS ^α =T. Assume that m is a positive integer.G is called anm-Cl-group if every subsetS ofG withS =S ⁻¹ and | S | ≤m is Cl. In this paper we prove that the alternating groupA ₅ is a 4-Cl-group, which was a conjecture posed by Li and Praeger.     

On a combinatorial problem in group theory

We say that a groupG ∈DS if for some integerm, all subsetsX ofG of sizem satisfy |X ²|<|X|², whereX ²={xy|x,y ∈X}. It is shown, using a previous result of Peter Neumann, thatG ∈DS if and only if either the subgroup ofG generated by the squares of elements ofG is finite, orG contains a normal abelian subgroup of finite index, on which each element ofG acts by conjugation either as the identity automorphism or as the inverting automorphism. Dedicated to John G. Thompson, the Wolf Prize Laureate in Mathematics for 1992 The first author wishes to thank the Department of Mathematics in the University of Napoli for their hospitality during the preparation of this paper.     

Some remarks on subgroups determined by certain ideals in integral group rings

LetG be a group,ZG the integral group ring ofG andI(G) its augmentation ideal. Subgroups determined by certain ideals ofZG contained inI(G) are identified. For example, whenG=HK, whereH, K are normal subgroups ofG andH∩K⊆ζ(H), then the subgroups ofG determined byI(G)I(H)I(G), andI ³(G)I(H) are obtained. The subgroups of any groupG with normal subgroupH determined by (i)I ²(G)I(H)+I(G)I(H)I(G)+I(H)I²(G), whenH′⊆[H,G,G] and (ii)I(G)I(H)I(G) when degH ²(G/H′, T)≤1, are computed. the subgroup ofG determined byI ⁿ(G)+I(G)I(H) whenH is a normal subgroup ofG withG/H free Abelian is also obtained     

The theory of stationary point processes

An axiomatic formulation is presented for point processes which may be interpreted as ordered sequences of points randomly located on the real line. Such concepts as forward recurrence times and number of points in intervals are defined and related in set-theoretic Note that for α∈A,G ^α may not coverG _α as a convex subgroup and so we cannot use Theorem 1.1 to prove this result. Moreover, all that we know about theG ^α/G_α is that each is an extension of a trivially ordered subgroup by a subgroup ofR. ItB is a plenary subset ofA, then there exists av-isomorphism μ ofG intoV(B, G ^β/G_β), but whether or not μ is ano-isomorphism is not known. Presently at Massachusetts Institute of Technology, Lexington, Mass., U.S.A. This work was supported by the National Aeronautics and Space Administration under research grant NsG-2-59     

Confined subgroups in periodic simple finitary linear groups

A subgroupX of the locally finite groupG is said to beconfined, if there exists a finite subgroupF≤G such thatX ^g∩F≠1 for allg∈G. Since there seems to be a certain correspondence between proper confined subgroups inG and non-trivial ideals in the complex group algebra ℂG, we determine the confined subgroups of periodic simple finitary linear groups in this paper. Dedicated to the memory of our friend and collaborator Richard E. Phillips     

Profinite groups with nonabelian crowns of bounded rank and their probabilistic zeta function

It has been conjectured by Mann that the infinite sum Σ _H μ(H,G)/|G:H| ^s , where H ranges over all open subgroups of a finitely generated profinite group G, converges absolutely in some half right plane if G is positively finitely generated. We prove that the conjecture is true if the nonabelian crowns of G have bounded rank. In particular Mann’s conjecture holds if G has polynomial subgroup growth or is an adelic profinite group.     

Additive Latin transversals and group rings

LetA={a ₁, …,a _k} and {b ₁, …,b _k} be two subsets of an abelian groupG, k≤|G|. Snevily conjectured that, when |G| is odd, there is a numbering of the elements ofB such thata _i+b _i,1≤i≤k are pairwise distinct. By using a polynomial method, Alon affirmed this conjecture for |G| prime, even whenA is a sequence ofk<|G| elements. With a new application of the polynomial method, Dasgupta, Károlyi, Serra and Szegedy extended Alon’s result to the groupsZ _p ^r andZ _p ^rin the casek<p and verified Snevily’s conjecture for every cyclic group. In this paper, by employing group rings as a tool, we prove that Alon’s result is true for any finite abelianp-group withk<√2p, and verify Snevily’s conjecture for every abelian group of odd order in the casek<√p, wherep is the smallest prime divisor of |G|. This work has been supported partly by NSFC grant number 19971058 and 10271080.     

A poisson formula for solvable Lie groups

Given a probability measure μ on a locally compact second countable groupG the space of bounded μ-harmonic functions can be identified withL ^∞(η, α) where (η, α) is a BorelG-space with a σ-finite quasiinvariant measure α. Our goal is to show that when μ is an arbitrary spread out probability measure on a connected solvable Lie groupG then the μ-boundary (η, α) is a contractive homogeneous space ofG. Our approach is based on a study of a class of strongly approximately transitive (SAT) actions ofG. A BorelG-space η with a σ-finite quasiinvariant measure α is called SAT if it admits a probability measurev≪α, such that for every Borel set A with α(A)≠0 and every ε>0 there existsg∈G with ν(gA)>1−ε. Every μ-boundary is a standard SATG-space. We show that for a connected solvable Lie group every standard SATG-space is transitive, characterize subgroupsH⊆G such that the homogeneous spaceG/H is SAT, and establish that the following conditions are equivalent forG/H: (a)G/H is SAT; (b)G/H is contractive; (c)G/H is an equivariant image of a μ-boundary.     

Relating a Locally Compact Abelian Group to Its Bohr Compactification

We find several conditions on a locally compact Abelian groupGnecessary and sufficient thatG⁺, the groupGin the topology inherited from its Bohr compactification, is realcompact. We show further for suchGthat every continuous real-valued function from a closed subgroup ofG⁺extends continuously overG⁺; when in additionGis discrete, every continuous function from a subgroup ofG⁺to a complete metric space extends continuously overG⁺. These results respond to several of the questions posed by E. K. van Douwen [Topology and Its Applications34(1990), 69–91].     

The groups of the semigroup of compact subsets of a topological group

The compact subsets of a topological groupG form a semigroup,S(G), when multiplication is defined by set product. This semigroup is a topological semigroup when given the Vietoris topology. It would be expected that the subgroups ofS(G) should in some way be related to the groupG. This is the case. It is shown that the subgroups ofS(G) are both algebraically and topologically exactly the groups obtained as quotients of certain subgroups ofG. One consequence of this is that every subgroup ofS(G) is a topological group. Conditions are also given for these subgroups to be open or closed. Green's relations inS(G) have a particularly nice formulation. As a result, the relationsD andJ are equal inS(G). Moreover, the Schützenberger group of aD-class is a topological group that is topologically isomorphic to a quotient of certain subgroups ofG.     

Division algebras with a projective basis

Letk be any field andG a finite group. Given a cohomology class α∈H ²(G,k ^*), whereG acts trivially onk ^*, one constructs the twisted group algebrak ^αG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Z _n×Z_n). This paper has two main results: First we prove that ifD=k ^α G is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and , the division algebraD=k ^α G is a product of cyclic algebras. Furthermore, ifD _p is ap-primary factor ofD, thenD _p is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and , the same result holds forD _p, p odd. Ifp=2 we show thatD ₂ is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z ₂×Z₂n.     

<Emphasis Type="Italic">G</Emphasis>-covering subgroup systems for the classes of supersoluble and nilpotent groups

LetF be a class of groups andG a group. We call a set Σ of subgroups ofG aG-covering subgroup system for the classF (or directly aF-covering subgroup system ofG) ifG ∈F whenever every subgroup in Σ is inF. In this paper, we provide some nontrivial sets of subgroups of a finite groupG which are simultaneouslyG-covering subgroup systems for the classes of supersoluble and nilpotent groups. Research of the first author is supported by the NNSF of China (Grant No. 10171086) and QLGCF of Jiangsu Province and a Croucher Fellowship of Hong Kong. Research of the second author is partially supported by a UGC (HK) grant #2060176 (2001/2002).     

Polynomial mappings of groups

A mapping ϕ of a groupG to a groupF is said to be polynomial if it trivializes after several consecutive applications of operatorsD _h ,h ∈G, defined byD _h ϕ(g)=ϕ(g) ⁻¹ ϕ(gh). We study polynomial mappings of groups, mainly to nilpotent groups. In particular, we prove that polynomial mappings to a nilpotent group form a group with respect to the elementwise multiplication, and that any polynomial mappingG→F to a nilpotent groupF splits into a homomorphismG→G’ to a nilpotent groupG’ and a polynomial mappingG’→F. We apply the obtained results to prove the existence of the compact/weak mixing decomposition of a Hilbert space under a unitary polynomial action of a finitely generated nilpotent group. This work was supported by NSF, Grants DMS-9706057 and 0070566.     

Primitive permutation groups of simple diagonal type

LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT ^m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT ^m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic. GivenT ^m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal subgroups are abstractly isomorphic toT ^m. Secondly, form the direct productS _m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS _m×OutT whose projections inS _m are primitive. The second result of the paper is that there is a bijection between these two sets. The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which can fall into a single abstract isomorphism class.     

C-Normality of Groups and Its Properties

A subgroupHis calledc-normal in groupGif there exists a normal subgroupNofGsuch thatHN=GandH∩N≤H_G, where[formula]is the maximal normal subgroup ofGwhich is contained inH. We obtain some results about thec-normal subgroups and use them to determine the structures of some groups.     

On galois groups and their maximal 2-subgroups

LetG be a finite group of even order, having a central element of order 2 which we denote by −1. IfG is a 2-group, letG be a maximal subgroup ofG containing −1, otherwise letG be a 2-Sylow subgroup ofG. LetH=G/{±1} andH=G/{±1}. Suppose there exists a regular extensionL ₁ of ℚ(T) with Galois groupG. LetL be the subfield ofL ₁ fixed byH. We make the hypothesis thatL ₁ admits a quadratic extensionL ₂ which is Galois overL of Galois groupG. IfG is not a 2-group we show thatL ₁ then admits a quadratic extension which is Galois over ℚ(T) of Galois groupG and which can be given explicitly in terms ofL ₂. IfG is a 2-group, we show that there exists an element α ε ℚ(T) such thatL ₁ admits a quadratic extension which is Galois over ℚ(T) of Galois groupG if and only if the cyclic algebra (L/ℚ(T).a) splits. As an application of these results we explicitly construct several 2-groups as Galois groups of regular extensions of ℚ(T).     

Some intersections and identifications in integral group rings

LetZG be the integral group ring of a groupG and I(G) its augmentation ideal. For a free groupF andR a normal subgroup ofF, the intersectionI ⁿ⁺¹ (F) ∩I ⁿ⁺¹ (R) is determined for alln≥ 1. The subgroupsF ∩ (1+ZFI (R) I (F) I (S)) ANDF ∩ (1 + I (R)I ³ (F)) of F are identified whenR and S are arbitrary subgroups ofF.     

Quasinormalität in topologischen Gruppen

LetQ be a subgroup of the locally compact groupG. Q is called a topologically quasinormal subgroup ofG, ifQ is closed and for each closed subgroupA ofG. We prove: If the compact elements ofG form a proper subgroup, compact topologically quasinormal subgroups ofG are subnormal of defect 2. IfG is connected, compact topologically quasinormal subgroups ofG are normal. IfG/G ₀ is compact, connected topologically quasinormal subgroups ofG are normal.     

Permutability of minimal subgroups and<Emphasis Type="Italic">p</Emphasis>-nilpotency of finite groups

In this paper it is proved that ifp is a prime dividing the order of a groupG with (|G|,p − 1) = 1 andP a Sylowp-subgroup ofG, thenG isp-nilpotent if every subgroup ofP ∩G ^N of orderp is permutable inN _G (P) and whenp = 2 either every cyclic subgroup ofP ∩G ^N of order 4 is permutable inN _G (P) orP is quaternion-free. Some applications of this result are given. The research of the first author is supported by a grant of Shanxi University and a research grant of Shanxi Province, PR China. The research of the second author is partially supported by a UGC(HK) grant #2160126 (1999/2000).